I want to discuss some of the math materials that I've seen while subbing this week. Yesterday, I mentioned that I was in an integrated math class. But now one might wonder, since most textbooks in this country aren't integrated, what text do the students in this class use?

Well, apparently the students learn out of various packets. The packet I saw today was published by the Center of Mathematics and Teaching, Inc., and titled

*Mathlinks*. It is labeled 8-5, which stands for the fifth packet for the eighth grade, and the subject is "Expressions and Equations (Part) 2."

The first thing that we notice is that even though this is a ninth grade class, the packet is apparently for

*eighth*graders. Not only that, but the back of the packet lists the Common Core Standards included within, and many of them are for sixth and seventh grades. The justification for having so many earlier standards is that it is "Review of content essential for success in 8th grade":

CCSS.MATH.CONTENT.6.EE.A.3

Apply the properties of operations to generate equivalent expressions.

Apply the properties of operations to generate equivalent expressions.

*For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y*.
CCSS.MATH.CONTENT.6.EE.A.4

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

*For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.*.#### Reason about and solve one-variable equations and inequalities.

CCSS.MATH.CONTENT.6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?~~Use substitution to determine whether a given number in a specified set makes an equation or inequality true.~~

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?

CCSS.MATH.CONTENT.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.MATH.CONTENT.6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations of the form

Solve real-world and mathematical problems by writing and solving equations of the form

*x*+*p*=*q*and*px*=*q*for cases in which*p*,*q*and*x*are all nonnegative rational numbers.
CCSS.MATH.CONTENT.7.EE.B.4.A

Solve word problems leading to equations of the form

Solve word problems leading to equations of the form

*px*+*q*=*r*and*p*(*x*+*q*) =*r*, where*p*,*q*, and*r*are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.*For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?*
CCSS.MATH.CONTENT.8.EE.C.7.A

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form

*x*=*a*,*a*=*a*, or*a*=*b*results (where*a*and*b*are different numbers).
CCSS.MATH.CONTENT.8.EE.C.7.B

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

I wonder whether the students will start using ninth grade packets later in the year. If so, then here we see one advantage to integrated math -- it's easier to slow down for the students who may be below grade level just by giving them eighth-grade packets. If a class needs to accelerate, then the teacher can pass out

*tenth*-grade packets. This would be awkward under the traditional pathway where the class is Algebra I -- a group that wants to accelerate would have trouble because the next class is Geometry, which won't help them pass the PARCC End-of-Course Algebra I exam. (Notice that such would be easier under SBAC, since there is no End-of-Course exam.)
Thus, even though many Common Core/integrated math opponents are fond of tracking, we see that tracking, accelerating, and decelerating are actually

*easier*under integrated math.
I've discussed my ideal standards for Grades K-3. So far, I haven't discussed the higher grades, even though these are the grades that I will be teaching. Part of this is because I am still trying to decide whether I would have traditional or integrated math. Perhaps I'm leaning towards integrated math, but in a way that is more palatable to traditionalists.

For example, since many traditionalists like the Singapore standards, perhaps I would create an integrated course almost verbatim from the Singapore standards. Many traditionalists say that the eighth grade course should be Algebra I -- since this class prepares them for calculus in senior year and hence success at an Ivy League school or a STEM career.

So suppose the eighth grade course were Singapore's Secondary Two course. (Notice that Secondary Two is our eighth grade, since the last elementary grade is Primary Six or sixth grade.) Looking back at the Singapore link from yesterday (under O Level, for "ordinary level"), we see that much of what appears under "Numbers and Algebra" is already part of Algebra I, but the other two topics, "Geometry and Measurement" and "Statistics and Probability" don't appear in our Algebra I class, but are already part of the Common Core standards. Indeed, for Geometry, notice the three topics for Singapore -- congruence/similarity, the Pythagorean Theorem, and volumes of cones/spheres -- are nearly

*identical*to the three strands under the Common Core eighth grade Geometry domain.
The integrated math packet, if we compared it to the Singapore standards, appears to be written mostly at a Secondary One level -- this is to be expected, since the back of the packet admits that most of the Common Core Standards included are sixth or seventh grade standards. I look forward to being in an integrated classroom and reading the packets when the classes reach a geometry unit.

Here is a review worksheet for a quiz. It covers the first three sections of Chapter 6 of the U of Chicago text (including Section 6-3 on rotations, since I did mention them earlier this week).

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